Global Cauchy problem for semilinear hyperbolic systems with nonlocal interactions. Applications to Dirac equations

Abstract

We investigate the global Cauchy problem for a class of semilinear hyperbolic systems where the interaction can be nonlocal in space and time. We establish global existence theorems for the initial value problem when the nonlinearity is dissipative in a weak sense, and satisfies the causality condition. The argument is abstract and the technique is based on the nonlinear resolvent. We apply these results to get low regularity global solutions of several models for relativistic field theory: the Dirac–Maxwell–Klein–Gordon system, and the Thirring model on the Minkowski space–time R 1 + 1 ; the Dirac–Klein–Gordon system on Schwarzschild type manifolds, or outside a star undergoing a gravitational collapse to a Black-Hole.